1. Field of the Invention
The present invention relates to an information processing apparatus and method, and a program. More particularly, the present invention relates to an information processing apparatus which is capable of generating an image signal (with a sharply rising signal edge) at a resolution higher than in the related art at the time of high density (zoom) conversion and in which ringing is reduced, a method for use therewith, and a program for use therewith.
2. Description of the Related Art
Hitherto, there have been demands for converting an image of a standard resolution or a low resolution (hereinafter, referred to as an SD image as appropriate) into an image of a high resolution (hereinafter referred to as an HD image as appropriate) or for expanding such an image. In order to respond to such demands, hitherto, pixel values of missing pixels have been interpolated (compensated for) using so-called interpolation filters.
However, even if pixels are interpolated using interpolation filters, it is difficult to restore components that are not contained in an SD image, that is, components (high frequency components) of an HD image, presenting a problem that it is difficult to obtain an image of a high resolution.
Accordingly, in order to solve such problems, the present applicant of the present invention previously proposed a classification adaptive process (see, for example, Japanese Unexamined Patent Application Publication No. 2002-218413). A classification adaptive process is signal processing in which, by computing a linear primary expression (prediction computation expression) between pixel values of a plurality of pixels in a predetermined area of an input image and a group of coefficients that have been determined in advance by learning using a teacher image and a student image, an output image of high resolution is determined.
That is, if a classification adaptive process is applied to the above-described demands, linear association between an SD image and predetermined prediction coefficients implements an adaptive process for determining prediction values of pixels of an HD image. Such an adaptive process will be described further.
For example, it is now considered that prediction values E[y] of pixel values y of pixels forming an HD image (hereinafter referred to as HD pixels as appropriate) are determined as follows. That is, the following case will be considered. A prediction value E[y] is determined by using a linear primary association model that is defined by linear association between pixel values (hereinafter referred to as learning data as appropriate) x1, x2, . . . of several SD pixels (pixels forming an SD image with respect to an HD pixel will be referred to in this manner as appropriate), and predetermined prediction coefficients w1, w2, . . . .
In this case, the prediction value E[y] can be represented as in the following Expression (1).E[y]=w1x1+w2x2+ . . .  (1)
Therefore, in order to generalize Expression (1), a matrix X formed of a set of learning data is defined as in the following Expression (2). A matrix W formed of a set of prediction coefficients w is defined as in the following Expression (3). A matrix Y′ formed of a set of prediction values E[y] is defined as in the following Expression (4).
                    X        =                  [                                                                      X                  11                                                                              X                  12                                                            …                                                              X                                      1                    ⁢                                                                                  ⁢                    n                                                                                                                        X                  21                                                                              X                  22                                                            …                                                              X                                      2                    ⁢                                                                                  ⁢                    n                                                                                                      …                                            …                                            …                                            …                                                                                      X                                      m                    ⁢                                                                                  ⁢                    1                                                                                                X                                      m                    ⁢                                                                                  ⁢                    2                                                                              …                                                              X                  mn                                                              ]                                    (        2        )                                W        =                  [                                                                      W                  1                                                                                                      W                  2                                                                                    …                                                                                      W                  n                                                              ]                                    (        3        )                                          Y          ′                =                  [                                                                      E                  ⁡                                      (                                          y                      1                                        )                                                                                                                        E                  ⁡                                      (                                          y                      2                                        )                                                                                                      …                                                                                      E                  ⁡                                      (                                          y                      m                                        )                                                                                ]                                    (        4        )            
As a result of being defined as described above, an observation Expression of the following Expression (5) holds.XW=Y′  (5)
Prediction values E[y] close to pixel values y of HD pixels are determined by applying a least squares method to the observation Expression of Expression (5). In this case, a matrix Y formed of a set of true pixel values y of HD pixels serving as teacher data will be defined as in the following Expression (6). Furthermore, a matrix E formed of a set of remainders e of the prediction values E[y] with respect to the pixel values y of the HD pixels is defined as in the following Expression (7).
                    Y        =                  [                                                                      y                  1                                                                                                      y                  2                                                                                    …                                                                                      y                  m                                                              ]                                    (        6        )                                E        =                  [                                                                      e                  1                                                                                                      e                  2                                                                                    …                                                                                      e                  m                                                              ]                                    (        7        )            
In this case, a remainder expression of the following Expression (8) holds on the basis of Expression (5).XW=Y+E  (8)
In this case, a prediction coefficient wi for determining prediction values E[y] close to the pixel values y of the HD pixels can be determined by, for example, minimizing the squared error shown by the following Expression (9).
                              ∑                      i            =            1                    m                ⁢                  e          i          2                                    (        9        )            
Therefore, a case in which the squared error of Expression (9) is differentiated using a prediction coefficient wi, that is, a prediction coefficient wi that satisfies the following Expression (10), will be an optimum value for determining the prediction values E[y] close to the pixel values y of the HD pixels.
                                                        e              1                        ⁢                                          ∂                                  e                  1                                                            ∂                                  W                  i                                                              +                                    e              2                        ⁢                                          ∂                                  e                  2                                                            ∂                                  W                  i                                                              +          …          +                                    e              m                        ⁢                                          ∂                                  e                  m                                                            ∂                                  W                  i                                                                    =                  0          ⁢                      (                                          i                =                1                            ,              2              ,              …              ⁢                                                          ,              n                        )                                              (        10        )            
Therefore, first, by differentiating Expression (8) using the prediction coefficient wi, the following Expressions (11) holds.
                                                        ∂                              e                i                                                    ∂                              W                i                                              =                      x                          i              ⁢                                                          ⁢              1                                      ,                                            ∂                              e                i                                                    ∂                              W                i                                              =                      x                          i              ⁢                                                          ⁢              2                                      ,        …        ⁢                                  ,                                            ∂                              e                i                                                    ∂                              W                i                                              =                                    x              in                        ⁡                          (                                                i                  =                  1                                ,                2                ,                …                ⁢                                                                  ,                m                            )                                                          (        11        )            
On the basis of Expressions (10) and (11), the following Expressions (12) is obtained.
                                                        ∑                              i                =                1                            m                        ⁢                                          e                i                            ⁢                              x                                  i                  ⁢                                                                          ⁢                  1                                                              =          0                ,                                            ∑                              i                =                1                            m                        ⁢                                          e                i                            ⁢                              x                                  i                  ⁢                                                                          ⁢                  2                                                              =          0                ,        …        ⁢                                  ,                                            ∑                              i                =                1                            m                        ⁢                                          e                i                            ⁢                              x                in                                              =          0                                    (        12        )            
Furthermore, when the relationship among the learning data x, the prediction coefficient w, the teacher data y, and the remainder e in the remainder expression of Expression (8) is considered, the following normal Expressions (13) can be obtained on the basis of Expression (12).(Σi=1mxi1xi1)w1+(Σi=1mxi1xi2)w2+ . . . +(Σi=1mxi1xin)wn=(Σi=1mxi1yi)(Σi=1mxi2xi1)w1+(Σi=1mxi2xi2)w2+ . . . +(Σi=1mxi2xin)wn=(Σi=1mxi2yi). . .(Σi=1mxinxi1)w1+(Σi=1mxinxi2)w2+ . . . +(Σi=1mxinxin)wn=(Σi=1mxinyi)  i. (13)
It is possible to formulate the normal expressions of Expressions (13), the number of Expressions being equal to the number of prediction coefficients w to be determined. As a consequence, it is possible to determine an optimum prediction coefficient w by solving Expressions (13). However, to solve Expressions (13), in Expressions (13), it is necessary that the matrix formed of coefficients applied to the prediction coefficient w be regular. When Expressions (13) are to be solved, for example, a sweeping-out method (a Gauss-Jordan's elimination method) or the like can be applied.
An adaptive process is such that, in the manner described above, the optimum prediction coefficient w is determined in advance, and the prediction values E[y] close to the pixel values y of the HD pixels are determined in accordance with Expression (1) by using the prediction coefficient w.
The adaptive process differs from an interpolation process in that components that are not contained in the SD image, that is, components contained in the HD image, are reproduced. That is, the adaptive process is viewed to be the same as an interpolation process using a so-called interpolation filter as long as only Expression (1) is viewed. However, since the prediction coefficient w corresponding to the tap coefficient of the interpolation filter is determined by learning using teacher data y, the components contained in the HD image can be reproduced. That is, an image having a high resolution can easily be obtained. On the basis of the above, the adaptive process may be said to be a process having a function of creating (the resolution) of the image.